Properties of Multiplication of Integers – Chapter 1

CBSE CLASS 7 MATHEMATICS
INTEGERS – Chapter 1
Properties of Multiplication of Integers:

I. Closure under Multiplication:
For all integers a and b, a x b is an integer.
Consider two integers (-20) and (-5).
Their product = (-20) x (-5) = 100, which is an integer.
This shows that the product of two integers is again an integer. So integers are closed under multiplication.

II. Commutativity of Multiplication:
For any two integers a and b, a x b = b x a.
Consider two integers 3 and (-4).
Then 3 x (-4) = -12 and (-4) x 3 = -12, both are equal.
This shows that multiplication is commutative for integers.

III. Multiplication by zero.
For any integer a, a x 0 = 0 x a = 0
Examples:
(-3) x 0 = 0
0 x (-4) = 0
This shows that the product of a negative integer and zero is zero.

IV. Multiplicative Identity:
For any integer a, a x 1 = 1 x a = a.
Examples:
(-3) x 1 = -3
1 x (-6) = -6
This shows that 1 is the multiplicative identity for integers.

V. Associativity for Multiplication:
For any three integers a, b and c,
(a x b) x c = a x (b x c).
Consider-3, -2 and 5.
Look at [(-3) x (-2)] x 5 and (-3) x [(-2) x5]
Case 1.
[(-3) x (-2)] x 5 = 6 x 5 = 30
Case 2.
(-3) x [(-2) x 5] = (-3) x (-10) = 30
So we get the same answer in both the cases.
So we can say that integers are associative for multiplication.

VI. Distributive property:
For any integers a, b and c
a x (b + c) = a x b + a x c
Consider the integers -2, 3 and 5.
Look at (-2) x (3 + 5) and (-2) x 3 + (-2) x 5.
Case 1
(-2) x (3+5) = (-2) x 8 = -16
Case2
(-2) x 3 + (-2) x 5 = (-6) + (-10) = -16.
So we get the same answer in both the cases.
So we can say that the distributivity of multiplication over addition is true for integers.